Question 7 of 10
The vertex form of the equation of a parabola is y= 2(x + 4)²-7.
What is the standard form of the equation?

Answer:

x^2 + x -12

Step-by-step explanation:

Here, we want to get the quadratic polynomial given its roots

Mathematically, we have it that;

x = 3 and x = -4

We can have that re-written before equating to zero as;

x-3 and x + 4

We can then proceed to multiply these two binomials

(x-3)(x + 4)

= x(x+ 4) -3(x + 4)

= x^2 + 4x -3x -12

= x^2 + x - 12

Answer:

k < -1 or k > 11

Step-by-step explanation:

Given quadratic equation:

\(3x^2-x+ k = kx-1\)

First, rearrange the given quadratic equation in standard form ax² + bx + c = 0:

\(\begin{aligned}3x^2-x+ k &= kx-1\\3x^2-x+ k-kx+1&=0-kx+1\\3x^2-x-kx+ k+1&=0\\3x^2-(1+k)x+ (k+1)&=0\end{aligned}\)

\(3x^2-(1+k)x+ (k+1)=0\)

Comparing this with the standard form, the coefficients a, b and c are:

\(\boxed{\begin{minipage}{7 cm}\underline{Discriminant}\\\\$\boxed{b^2-4ac}$ \quad when $ax^2+bx+c=0$\\\\when $b^2-4ac > 0 \implies$ two real roots.\\when $b^2-4ac=0 \implies$ one real root.\\when $b^2-4ac < 0 \implies$ no real roots.\\\end{minipage}}\)

If the quadratic equation has two distinct roots, its discriminant is positive.

\(b^2-4ac > 0\)

Substitute the values of a, b and c into the discriminant:

\((-1 - k)^2-4(3)(k+1) > 0\)

Simplify:

\((-1 - k)(-1-k)-12(k+1) > 0\)

\(1+2k+k^2-12k-12 > 0\)

\(k^2+2k-12k+1-12 > 0\)

\(k^2-10k-11 > 0\)

Factor the left side of the inequality:

\(k^2+k-11k-11 > 0\)

\(k(k+1)-11(k+1) > 0\)

\((k-11)(k-1) > 0\)

If we graph the quadratic k² - 10k - 11, it is a parabola that opens upwards (since its leading coefficient is positive), and crosses the x-axis at k = -1 and k = 11. Therefore, the curve will be positive (above the x-axis) either side of the x-intercepts, so when k < -1 or k > 11.

Therefore, the range of values of k for which the given quadratic equation has two distinct roots is:

\(\boxed{k < -1 \; \textsf{or} \;k > 11}\)

Answer:

2nd answer

Step-by-step explanation:

a triangles area is its height x base x half so 8x4/2 is 16

the vertex of the function g(x) = f(x 2) .

If a quadratic equation is expressed as the following

                          \(y=a(x-h)^{2}+k\)

It is then said to be in vertex form. Because the vertex point (peak point) occurs on the graph of this equation, it is so named (h,k)

The parabola that the quadratic equation depicts has this point (h,k) as its vertex.

We first remove the x squared coefficient, and then inside the bracket, we strive to create a condition that resembles a perfect square.

So, this is what we have:

                         \($f(x)=x^{2}+6 x+3$\\$g(x)=f(x-2)$\)

Thus, we obtain:

                        \(\begin{aligned}&g(x)=f(x-2) \\&g(x)=(x-2)^{2}+6(x-2)+3 \\&g(x)=x^{2}-4 x+4+6 x-12+3 \\&g(x)=x^{2}+2 x-5\end{aligned}\)

Vertex form transformation of g(x):

                             \(\begin{aligned}&g(x)=x^{2}+2 x-5 \\&g(x)=x^{2}+2 x+1-1-5 \\&g(x)=(x+1)^{2}-9 \\&g(x)=(x-(-1))^{2}-6 \\&g(x)=(x-(-1))^{2}+(-6)\end{aligned}\)

By comparing this\(a(x-h)^{2}+k\) to, we obtain the coordinates of the vertex of the graph of g(x) as (h,k) = (-1,-6), as shown below.

As a result, the vertex of the function g(x) = f(x 2) .

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What is h?

maybe 25%

i hope it helped you sorry if it didn't

Answer:

25%

Step-by-step explanation:

Answer:

A =338.56 units ^2

Step-by-step explanation:

The area  is now

A = (SF)^2 * s^2

where s is the side length and SF is the scale factor

A = (4)^2 (4.6)^2

A =338.56

Answer:

The critical value for a 95% confidence level is 1.711.

Step-by-step explanation:

a=0.05

1-a=0.95

Degrees of freedom df =21-1=20

using the t-distribution table, the critical value for a 95% confidence level and df=20 is 1.711

a) The value of k in the equation y = 80e^kx - 300x can be found by substituting the given values of x and y into the equation. The value of k is ln(880)/1080, where ln represents the natural logarithm.

b) To solve the equation log(7x + 5) - log(x - 5) = 1 + log3(x + 2) (x > 5), we can use logarithmic properties to simplify the equation and solve for x. The solution involves manipulating the logarithmic terms and applying algebraic techniques.

c) In Figure 4, given the information about the quadrilateral PQRS, we can find the value of x using the given lengths and angles. By applying trigonometric properties and solving equations involving angles, we can determine the value of x. Additionally, the size of angle SQR can be found by using the properties of triangles and angles.

a) Substituting the values x = 1 and y = 1080 into the equation y = 80e^kx - 300x, we have 1080 = 80e^(k*1) - 300*1. Solving for k, we get k = ln(880)/1080.

b) Manipulating the given equation log(7x + 5) - log(x - 5) = 1 + log3(x + 2), we can use the logarithmic property log(a) - log(b) = log(a/b) to simplify it to log((7x + 5)/(x - 5)) = 1 + log3(x + 2). Further simplifying, we get log((7x + 5)/(x - 5)) - log3(x + 2) = 1. Using logarithmic properties and algebraic techniques, we can solve this equation to find the value(s) of x.

c) In triangle PQS, we know the length of PS (13√2), angle PSQ (45°), and the area of triangle PQS (130 m²). Using the formula for the area of a triangle (Area = 0.5 * base * height), we can find the height PQ. In triangle SRQ, we know the length of SR (17), angle SRQ (64°), and the length SQ (x). By applying trigonometric ratios, such as sine and cosine, we can determine the values of x and angle SQR.

By following the steps outlined in the problem, the values of k, x, and angle SQR can be found, providing the solutions to the given equations and geometric problem.

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The equation of line p is y = (3/4)x + 13/2.

The equation of line q is y = (3/5)x + 23/5.

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

Line p passes through points (-2, 5) and (3, 8).
This means,

Slope = (8 - 5) / (3 + 2) = 3/4

m = 3/4

And,

Consider (-2, 5) = (x, y)

5 = (3/4) x -2 + c

5 = -3/2 + c

c = 5 + 3/2

c = 13/2

So,

y = (3/4)x + 13/2

Line q passes through the points (-1, 4) and (4, 7).

Slope = (7 - 4) / (4 + 1) = 3/5

m = 3/5

And,

Consider (-1, 4) = (x, y)

4 = (3/5) x (-1) + c

4 = -3/5 + c

c = 4 + 3/5

c = 23/5

So,

y = (3/5)x + 23/5

Thus,

The equation of line p is y = (3/4)x + 13/2.

The equation of line q is y = (3/5)x + 23/5.

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The complete question:

Find the equation of the Line p passes that passes through points (-2, 5) and (3, 8) and Line q that passes through points (-1, 4) and (4, 7).

it is not possible for det A to equal 3 if the entries of both A and A^-1 are integers.

The determinant of a matrix and its inverse are multiplicative inverses of each other, meaning that det(A)det(A^-1) = 1. If det A = 3, then det(A^-1) = 1/3. However, since the entries of both A and A^-1 are integers, this is a contradiction, as the determinant of a matrix with integer entries must also be an integer. Therefore, it is impossible for det A to equal 3 in this scenario.

it explains the reasoning behind the solution and provides a deeper understanding of the concept of determinants.

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The cost of the lawn fertilizing services before the 30% discount was applied was $651.43.

Let's assume the cost of the services before the discount is x dollars. Since Mark received a 30% discount, he paid 70% of the original cost after the discount. We can represent this mathematically as:

0.70x = $456

To find the value of x, we can divide both sides of the equation by 0.70:

x = $456 / 0.70 ≈ $651.43

Therefore, the cost of the services before the discount was applied is approximately $651.43.

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Answer:

a) c, s

b) P(c, s) = 60c + 90s

c) c + s ≤ 320

50c + 100s ≤ 20000

100c + 40s ≤ 19200

d) \(c + s+s_1=320\)

\(50c + 100s+s_2=20000\)

\(100c + 40s+s_3=19200\)

Step-by-step explanation:

a) Let c represent the acres of corn planted and s represent the acres of soybeans planted.

the decision variables of this problem are c and s

a) Each acre of corn yields a profit of $60 and Each acre of soybeans yields a profit of $90

The objective function P(c, s) = 60c + 90s

c)

1) The farmer has a 320 acre farm on which she plants two crops: corn and soya beans. The constraint for this problem is:

c + s ≤ 320

2)  her expenses are $50 and for each acre of soybeans planted, he expenses are $100. the farmer has only $20,000 on hand. The constraint is:

50c + 100s ≤ 20000

3) Each acre of corn requires 100 bushels of storage and each acre of soybeans requires 40 bushels of storage.  the total amount of storage space available is 19,200 bushels, The constraint is:

100c + 40s ≤ 19200

d) We have a total of 3 equations. To balance the equations,we would introduce 4 slack variables: \(s_1,s_2,s_3\)

\(c + s+s_1=320\)

\(50c + 100s+s_2=20000\)

\(100c + 40s+s_3=19200\)

Therefore solving these equations gives

c=140, s=130

P(c, s) = 60c + 90s = 60(140) + 90(130)= $20100

The required number of seats in row 20 is 126.

In this question, we are given that the number of seats per row increases at a constant rate on the main floor of the Kodak Hall at the Eastman Theater.

Steven counts 31 seats in row 3 and 37 seats in row 6. We need to determine how many seats are there in row 20. Let the number of seats in row 3 be a and the common difference be d.

Then, we can express the number of seats in row 6 and row 20 in terms of a and d.

The number of seats in row 6 is given to be 37. So, using the formula for the nth term of an arithmetic sequence, we can write an = a + (n - 1)d37 = a + 3d ...(1)Similarly, the number of seats in row 3 is given to be 31.

So, we have a3 = a + 2d31 = a + 2d ...(2)Subtracting equation (2) from equation (1), we get:37 - 31 = (a + 3d) - (a + 2d)6 = Thus, the common difference is 6. Now, using equation (2), we can write:31 = a + 2d31 = a + 2(6)31 = a + 123 = Therefore, the number of seats in row 20 will be:a20 = a + 19d = 12 + 19(6) = 126. So, there are 126 seats in row 20.

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Given the expression: (3.2+0.2)2+12​

To get the value of the expression, do the following:

1. add 3.2 and 0.2

2. multiply 3.4 by 2

3. add 6.8 and 12

So, the value of the expression = 18.8

The given statement  "The Puritan colony of Massachusetts Bay was not known for its high levels of religious toleration." is False because, In fact, the Puritans who founded the colony in the early 17th century were known for their strict religious beliefs and practices.

They came to the New World seeking to establish a "city upon a hill" that would serve as a shining example of Christian virtue and piety. As a result, they were deeply suspicious of anyone who did not share their beliefs and sought to create a society that was strictly controlled by the church.

One of the most famous examples of the lack of religious tolerance in Massachusetts Bay was the case of Anne Hutchinson. Hutchinson was a Puritan woman who held religious meetings in her home where she preached her own interpretations of scripture. Her views were considered heretical by the Puritan leadership, and she was put on trial and ultimately banished from the colony.

Similarly, the Puritans were hostile to Quakers and other religious groups that they saw as a threat to their way of life. Quakers were often subjected to harsh punishments such as public whippings and banishment.

In short, while the Puritans of Massachusetts Bay may have believed in the importance of religious freedom, they did not practice it in a way that we would recognize today. Their society was highly regulated and tightly controlled by the church, and dissenters were not tolerated.

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If the product of 14 and a number x is -75, then the number is -5.36

Consider the unknown number as x

The product of 14 and a number

Then the expression will be

14x

The product of 14 and a number x is -75

The equation will be

14x = -75

The equation is the mathematical statement that shows the relationship between two expression with an equal sign

Divide both side of the equation by 14

14x / 14 = -75 / 14

x = -75 / 14

x = -5.36

Hence, if the product of 14 and a number x is -75, then the number is -5.36

The complete question is:

The product of 14 and a number x is -75, find the value of x.

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The derivative \(\( y' \)\) of the implicit function \(\( y^2 - xy = -2 \)\) is 0, indicating a constant slope with no change in relation to \(\( x \)\).

To find \(\( y' \)\)for the implicit function \(\( y^2 - xy = -2 \)\), we can differentiate both sides of the equation with respect to \(\( x \)\) using the chain rule. Let's go step by step:

Differentiating \(\( y^2 \)\) with respect to \(\( x \)\) using the chain rule:

\(\[\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}\]\)

Differentiating \(\( xy \)\) with respect to \(\( x \)\) using the product rule:

\(\[\frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + y \cdot \frac{dx}{dx} = x \cdot \frac{dy}{dx} + y\]\)

Differentiating the constant term (-2) with respect to \(\( x \)\) gives us zero since it's a constant.

So, the differentiation of the entire equation is:

\(\[2y \cdot \frac{dy}{dx} - (x \cdot \frac{dy}{dx} + y) = 0\]\)

Now, let's rearrange the terms:

\(\[(2y - y) \cdot \frac{dy}{dx} - x \cdot \frac{dy}{dx} = 0\]\)

Simplifying further:

\(\[y \cdot \frac{dy}{dx}\) \(- x \cdot \frac{dy}{dx} = 0\]\)

Factoring out:

\(\[(\frac{dy}{dx})(y - x) = 0 \]\)

Finally, solving:

\(\[\frac{dy}{dx} = \frac{0}{y - x} = 0\]\)

Therefore, the derivative \(\( y' \)\) of the given implicit function is 0.

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Answer:

Let h = the number of hamburgers

Let d = the number of drinks

System of Equations:

5.50h + 2.25d = 42

h + d = 10

Step-by-step explanation:

For our system of equations, we can allow h to represent the number of hamburgers purchased and d to represent the number of drinks purchased.

First equation:  We know that (cost of one hamburger * quantity of one hamburger) + (cost of one drink * quantity of one drink) = total cost

Since it costs $5.50 per hamburger, $2.25 per drink, and the total cost is $42.00, our first equation is:  

5.50h + 2.25d = 42

Second equation:  We further know that total quantity of hamburgers + total quantity of drinks = 10 amount of food and drinks purchase

Thus, our second equation is:

h + d = 10

The system of equations for the number of hamburgers (H) and drinks (D) is: 5.50H + 2.25D = 42 and H + D = 10.

Let's denote the number of hamburgers as H and the drinks as D. Hence, the system of equation can be written as follows:

1. 5.50H + 2.25D = 42, which represents the total cost of hamburgers and drinks.

2. H + D = 10, which represents the total number of items bought.

You can solve this system of equations by either substitution, elimination, or graphing methods to get the exact number of hamburgers and drinks.

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Step-by-step explanation:

circumference=πd

22/7×4

12.57

perimeter=12.57+9+9+2+6

= 38.57

area=6×9=54

1/2πr2

1/2×22/7×4×4

=25.14

54-25.14

area=28.86cm2

Answer:

The image is not clear but get the idea

Answer:

your question is empty please give right questions to get answers

Answer:

equal to 1

Step-by-step explanation:

I believe this is the answer. Also I see that ur new. Well welcome to Brainly.

Answer:

27.2368421053

Step-by-step explanation:

All parts of the problem given have been explained and provided below.

Variables in any particular dataset are defined as any characteristics, numbers, or quantity that can be measured or counted to give information about any object, on which the dataset is based.

So, the variables in this given dataset are;

Overall score

Recommended

owner satisfaction

Overall miles per gallon

Acceleration (0-60) sec

Categorical or qualitative variables are those which can't be quantified or measured. Their values are defined by an order relation between the different categories.

the categorical variables in this dataset are,

Recommended

Owner satisfaction

Quantitative variables are those which can be measured in an order to provide information about an object.

the quantitative variables in this dataset are,

Overall score

Overall miles per gallon

Acceleration (0-60) sec

In the given dataset, 7 vehicles are recommended,

So their percentage would be,

7/15 × 100 = 46.67%

The average overall miles per gallon of these 15 vehicles is 25.4.

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Answer:

Yes, Rain is correct

238.5 square inches

Step-by-step explanation:

Surface Area of a Triangular Pyramid,A = 1/2(a × b) + 3(1/2 × b × s)

= (1/2 × 10 × 8.7) + 3(1/2 × 10 × 13)

= 43.5 + 3(65)

= 43.5 + 195

= 238.5 square inches

The surface area of the triangular pyramid = 238.5 square inches

The surface area of the net of the triangular pyramid is 223.5 in². Rain is not correct.

An equation is an expression that shows the relationship between two or more numbers and variables.

From the net attached:

Surface area = 3(1/2 * 10 * 12) + (1/2 * 10 * 8.7)

Surface area = 223.5 in²

The surface area of the net of the triangular pyramid is 223.5 in². Rain is not correct.

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Answer:

l = 2w - 3

width = 7 meters

Step-by-step explanation:

The length of a rectangle is 3 meters less than twice the width. Part A: write an expression to find the length of the rectangle.

l = 2w - 3

_________________________________________

Part B: The length of the rectangle is 11 meters. What is the width of the rectangle?

when l = 11 then:

11 = 2w - 3

add 3 to to both sides:

11 + 3 = 2w - 3 + 3

14 = 2w

divide both sides by 2:

14/2 = 2w/2

w = 7

Answer:

7i

Step-by-step explanation:

√-49= √-1*7²= 7√-1= 7i

Answer:

7i

Step-by-step explanation:

I hope this was helpful.

The original number is 0.8.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.

Variables are the name given to these symbols because they lack set values. We frequently observe constant change in specific values in our day-to-day situations. But the need to depict these shifting values is ongoing.

In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.

Given:

let the number be x.

A certain number was multiplied by 5, then decreased by 3, then halved. The result was 0.3 less than the original number.

So, the mathematical expression is

1/2( 5x- 3) = x- 0.3

Now, solving for x

Multiply both side by 2

5x-3 = 2x- 0.6

5x- 2x= -0.6 + 3

3x= 2.4

Divide both side by 3

3x/3 = 2.4/3

x= 0.8

Hence, the original number is 0.8.

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